Hi Greenfoot,
A pruning rule of thumb is to remove no more than 1/3 of a tree per year.
So I came up with a three year plan for trees that I see as needing attention:
First year - identify new central leader. Prune the most obvious competition for light, in particular anything higher.
Second year, prune second third of competition.
Third year, prune last third of competition.
I would then pretty much follow Del's advice and wait a decade and see where the tree want to go.
Humid summers are common here so a common fruit tree form is more open in the center to minimize fungal diseases.
In dry areas like Colorado, a more dense crown can minimize moisture loss.
For specimen yard shade trees, best is probably to let the tree grow mostly how it would naturally which is a pyramid shape.
Many bird species like the seeds.
TheJollyLogger's comment,"Going forward, any newly established leader will have a weak attachment point though." is accurate. And with an established yard tree you might want to just accept the risk.
Thanks and good health, Weogo